'This Gmail user does not exist...'
This error usually occurs when there are typos in the recipient's email address. Some common errors include the following:
Quotation marks: or
Dots at the end of the address:
Spaces before or after an address: < username@gmail.com>,
If you're sure there are no mistakes, the Gmail address may have been disabled, deleted, or never existed in the first place.
Additional information for Google Apps
If you're using Google Apps and you're trying to send mail to somebody else in your domain, you can check that the recipient address is valid by searching for the address in your contacts.
The search results will include everybody in your domain unless your domain administrator has chosen to hide individual users from the directory, or has disabled contacts sharing entirely.
If you're a Google Apps administrator, you can search for the email address in your Admin console